If `y=4cos4x` find `int y dx`

Explanation

To solve the problem of finding the integral of \( y = 4 \cos(4x) \), we will follow these steps:

### Step 1: Set up the integral
We need to find the integral of \( y \) with respect to \( x \):
\[ I = \int y \, dx = \int 4 \cos(4x) \, dx \]

### Step 2: Factor out the constant
Since 4 is a constant, we can factor it out of the integral:
\[ I = 4 \int \cos(4x) \, dx \]

### Step 3: Integrate \( \cos(4x) \)
To integrate \( \cos(4x) \), we use the formula for the integral of \( \cos(ax) \), which is:
\[ \int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C \]
In our case, \( a = 4 \):
\[ \int \cos(4x) \, dx = \frac{1}{4} \sin(4x) + C \]

### Step 4: Substitute back into the integral
Now we substitute back into our integral:
\[ I = 4 \left( \frac{1}{4} \sin(4x) + C \right) \]

### Step 5: Simplify the expression
The \( 4 \) and \( \frac{1}{4} \) will cancel out:
\[ I = \sin(4x) + C \]

### Final Answer
Thus, the integral of \( y = 4 \cos(4x) \) with respect to \( x \) is:
\[ \int y \, dx = \sin(4x) + C \]